It may take O(1) (not really but I will play along) to "solve" a single problem instance after it has been mapped, but you still need to MAP the problem to the quantum solver for each instance. And the mapping process/problem is definitively not O(1), or even P for that matter, in terms of complexity. There are 3-sat solvers for example which compute solutions in P time, as long as the mapping process is ignored.

Which is why I said that the speed of the computer is irrelevant when trying to prove that P=NP.

I think that quantum computing cannot ever apply to proving P=NP or P!=NP because complexity theory is predicated on running programs on deterministic machines. Quantum machines would be non-deterministic.